Convex combination







In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the count of the weights as in a standard weighted average.

Formal definition
More formally, given a finite number of points $$x_1, x_2, \dots, x_n$$ in a real vector space, a convex combination of these points is a point of the form
 * $$\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n$$

where the real numbers $$\alpha_i$$ satisfy $$\alpha_i\ge 0 $$ and $$\alpha_1+\alpha_2+\cdots+\alpha_n=1.$$

As a particular example, every convex combination of two points lies on the line segment between the points.

A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations.

There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval $$[0,1]$$ is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).

Other objects

 * A random variable $$X$$ is said to have an $$n$$-component finite mixture distribution if its probability density function is a convex combination of $$n$$ so-called component densities.

Related constructions

 * A conical combination is a linear combination with nonnegative coefficients. When a point $$x$$ is to be used as the reference origin for defining displacement vectors, then $$x$$ is a convex combination of $$n$$ points $$x_1, x_2, \dots, x_n$$ if and only if the zero displacement is a non-trivial conical combination of their $$n$$ respective displacement vectors relative to $$x$$.
 * Weighted means are functionally the same as convex combinations, but they use a different notation. The coefficients (weights) in a weighted mean are not required to sum to 1; instead the weighted linear combination is explicitly divided by the sum of the weights.
 * Affine combinations are like convex combinations, but the coefficients are not required to be non-negative. Hence affine combinations are defined in vector spaces over any field.